The pioneering works of [1]}, [2]} were first to describe the development of "long" wavelength deformations on the surface of the film. They identified these deformations as hydrodynamic instability, and therefore called it H-mode instability. The Orr-Sommerfeld approach has been followed in much of the work devoted to studying the local stability of falling films with respect to the hydrodynamic instability. [3]} and [4]} were first to follow this approach as they performed a temporal stability analysis in which the wave velocity and amplification factor for a perturbation with a given wavelength is sought. They identified the onset of the H-mode for small wavenumbers as a function of the critical Reynolds number and inclination angle \(\beta \)  where \(Re_c = (5/6) \cot (\beta )\) . On the other hand, [5]} performed a spatial stability analysis where the spatial amplification and the wavelength of a disturbance with a prescribed frequency is sought. A more simplified approach was formulated by [6]}, where he used long wave expansion to obtain a temporal evolution equation (Benny's equation) that describes the interface development for small wavenumber perturbations. The evolution equation can be utilized to obtain a dispersion relation of the temporal amplification rate in the regions of small wavenumbers and weak inertia. Various physical effects were considered while studying the local stability of falling films such as heating and phase change [7]}, shearing gas [8]}, and flexible inclined surface [9]}, etc. For the wide range of mathematical approaches and physical effects considered while studying the interface evolution and local stability of falling films, refer to the reviews by [10]}, [11]}, and [12]}.
